Flip-pushdown automata are pushdown automata with an extra ability to reverse the contents of the pushdown store. A generalisation of the Pumping lemma for context-free languages is presented, which applies to the families of languages accepted by flip-pushdown automata with k pushdown flips, for an arbitrary constant k. The presented result gives rise to a new technique for disproving existence of flip-pushdown automata with a constant number of flips, which is significantly simpler compared to methods used for this purpose so far.